The Geometry and Topology of Compact Complex Manifolds with RC-Positive Tangent Bundles

. In this note, we give a brief exposition on the recent progress on the geometry and topology of compact complex manifolds with RC-positive tangent bundles, with particular focus on several open problems proposed by S.-T. Yau around 40 years ago. The uniformization theorem demonstrates satis-factorily that: every simply connected Riemann sur-face is isomorphic to one of three Riemann surfaces: the open unit disk D , the complex plane C , or the Riemann sphere S 2 . However, the geometry and topology of higher dimensional complex manifolds are far more complicated, and the classifica-tions of such manifolds are still challenging tasks in modern geometry. As analogous to the geometric properties of three model Riemann surfaces, many terminologies are developed to investigate higher dimensional complex manifolds. For instances, a vari-ety of curvature notions are used in differential geometry, e.g., holomorphic bisectional curvature, holomorphic sectional curvature, Ricci curvature and etc. With the help of the intersection theory, algebraic geometers created many algebraic concepts to de-scribe the geometry of algebraic manifolds, such as ampleness, nefness and so on. On the other hand, it is natural to investigate complex manifolds by ana-lyzing model Riemann surfaces contained in them. The category of algebraic manifolds containing rational curves ( CP 1 ∼ = S 2 ) plays a significant role in alge-* braic geometry, and complex manifolds without entire curves C —so called hyperbolic manifolds—are fundamental in analytical geometry.